SOME WEIGHTED SUMS OF POWERS OF FIBONACCI POLYNOMIALS

The sequence of generalized bivariate Fibonacci polynomials Gn (x, y) is defined by the second-order recurrence Gn+2 (x, y) = xGn+1 (x, y) + yGn (x, y), with arbitrary initial conditions G0 (x, y) and G1 (x, y). When G0 (x, y) = 0 and G1 (x, y) = 1 we have the bivariate Fibonacci polynomials Fn (x, y), and when G0 (x, y) = 2 and G1 (x, y) = x we have the bivariate Lucas polynomials Ln (x, y). (What we will use about these polynomials is contained in reference [2].) The corresponding extensions to negative indices are given by F n (x, y) = ( y) n Fn (x, y) and L n(x, y) = ( y) n Ln(x, y), n 2 N, respectively. In the case y = 1, we have the Fibonacci and Lucas polynomials (in the variable x), Fn (x, 1) and Ln (x, 1), denoted simply as Fn (x) and Ln (x). By setting x = 1 in these polynomials, we obtain the numerical sequences Fn (1) and Ln (1), denoted as Fn and Ln, corresponding to the Fibonacci sequence Fn = (0, 1, 1, 2, 3, 5, . . .) and the Lucas sequence Ln = (2, 1, 3, 4, 7, 11, . . .). In a recent work [7] we showed that the sequence Gtsn+μ (x, y), where t, s, k 2 N and μ 2 Z are given parameters, can be written as a linear combination of the (s-)Fibopolynomials n+tk i tk Fs(x,y) , i = 0, 1, . . . , tk, according to


Introduction
The sequence of generalized bivariate Fibonacci polynomials G n (x, y) is defined by the second-order recurrence G n+2 (x, y) = xG n+1 (x, y) + yG n (x, y), with arbitrary initial conditions G 0 (x, y) and G 1 (x, y).When G 0 (x, y) = 0 and G 1 (x, y) = 1 we have the bivariate Fibonacci polynomials F n (x, y), and when G 0 (x, y) = 2 and G 1 (x, y) = x we have the bivariate Lucas polynomials L n (x, y).(What we will use about these polynomials is contained in reference [2].)The corresponding extensions to negative indices are given by F n (x, y) = ( y) n F n (x, y) and L n (x, y) = ( y) n L n (x, y), n 2 N, respectively.In the case y = 1, we have the Fibonacci and Lucas polynomials (in the variable x), F n (x, 1) and L n (x, 1), denoted simply as F n (x) and L n (x).By setting x = 1 in these polynomials, we obtain the numerical sequences F n (1) and L n (1), denoted as F n and L n , corresponding to the Fibonacci sequence F n = (0, 1, 1, 2, 3, 5, . ..) and the Lucas sequence L n = (2, 1, 3, 4, 7, 11, . ..).
Recall that for integers n, p 0, we have n 0 Fs(x,y) = n n Fs(x,y) = 1 and ✓ n p (see [6]), and when x = y = s = 1, we have the (usual) Fibonomials n p F , introduced and studied by Hoggatt [3] in 1967.
If (x, y) is a (non-zero) given real function of the real variables x and y, we see from (1) that Expression (2) can be written as ⇥G k ts(i j)+µ (x, y) y sj(j 1) 2 i tk+q+m (x, y) This simple observation derived from (1) allows us to obtain closed formulas, in terms of Fibopolynomials, for the weighted sums P q n=0 n (x, y) G k tsn+µ (x, y), as the following proposition states (with a straightforward proof using (3)).
G k ts(i j)+µ (x, y) y sj(j 1) 2 then the weighted sum of k-th powers P q n=0 n (x, y) G k tsn+µ (x, y) can be expressed as a linear combination of the s-Fibopolynomials q+m tk Fs(x,y) , m = 1, 2, . . ., tk, according to ⇥G k ts(i j)+µ (x, y) y sj(j 1) 2 i tk+m (x, y) Moreover, suppose expression ( 5) is valid for some weight function (x, y).Then z = (x, y) is a root of (4).
Let us consider the simplest case t = k = 1.Equation ( 4) is in this case If G s+µ (x, y) L s (x, y) G µ (x, y) 6 = 0, we have from (6) the weight Expression (5) for the corresponding weighted sum is in this case q X n=0 n (x, y) G sn+µ (x, y) = G µ (x, y) q (x, y) In the Fibonacci case we have, from (7) and (8), that if µ 6 = s then Similarly, from ( 7) and (8), we have in the Lucas case that Thus, if µ 6 = 0, s we can combine (9) and (10) together as We call attention to the fact that the term 1 Fs(x,y) F s(q+1) (x, y) in (11) does not depend on the parameter µ.That is, in (11) we have infinitely many weighted sums of q+1 bivariate Fibonacci and Lucas polynomials that are equal to 1 Fs(x,y) F s(q+1) (x, y).We can have a slight generalization of (11) if we substitute x by L r (x, y) and y by ( 1) r+1 y r , where r 2 Z.By using the equations we get for r 6 = 0 (and µ 6 = 0, s in the Fibonacci sums) rs F r(µ s) (x, y) rs L r(µ s) (x, y) Some examples from (12) (corresponding to some specific values of µ) are the following: y) .
• With s = 2 we have y) .

Weighted Sums of Cubes
In this section we will obtain expressions for some weighted sums of cubes of Fibonacci polynomials.More specifically, we will consider the Fibonacci case of Proposition 1 when k = 3, µ = 0 and t = 1.
In this case, equation ( 4) is (after some simplifications) Thus, we have two weights, namely The corresponding weighted sum (5) can be written as where (x) is any of the weights (20).
In the case x = 1, we have in particular the following numerical identities (obtained by setting s = 1 and s = 2 in (21) with the corresponding weights (20)) We will show now some di↵erent versions of (21), involving Chebyshev polynomials of the first kind T n (x), or of the second kind U n (x).The results are the following.
Proposition 2. (a) If s is even, we have the following weighted sums of cubes of Fibonacci polynomials, valid for any integer l 0 ! .
(b) If s is odd, we have the following weighted sums of cubes of Fibonacci polynomials (where i 2 = 1), valid for any integer l 0 Proof.Recall that Chebyshev polynomials of the first kind T n (x) can be calculated as and that Chebyshev polynomials of the second kind U n (x) can be calculated as We can write (21) as where l is a non-negative integer.If s is even, the weights (20) are Thus, (24) follows from (28), ( 30) and (31).Similarly, (25) follows from (29), (30) and (31).
We note that the integer parameter l 0 gives us, in (24) and (25), infinitely many weighted sums of cubes of Fibonacci polynomials for each even s, and in (26) and (27) gives us infinitely many weighted sums of cubes of Fibonacci polynomials for each odd s.
We show some numerical examples from formulas (24) to (27).We set x = 1 and q = 5 in them.

Other Weighted Sums
In this section we will obtain expressions for some other weighted sums of Fibonacci and Lucas polynomials.More specifically, we will set µ = 0 in Proposition 1 and consider the following cases: Let us begin with the case (i).When G = F , equation ( 4) is and with Observe that where s L 2s (x) z + 1 is a factor of the right-hand sides of ( 35) and (36), we have for both, the Fibonacci and the Lucas cases, the following weights In the Fibonacci case we have in addition the weight (x) = 1 (from the factor z + 1 of the right-hand side of (35)).In the Lucas case we have in addition the following weights (from the factor L 2 2s (x) z 2 (3L 4s (x) + 2) z + 4 of the right-hand side of (36)) In the Fibonacci case the corresponding sum ( 5) is , (where (x) is any of the weights (37) or (x) = 1).In the Lucas case the sum ( 5) is + ( 1) , (where (x) is any of the weights (37) or ( 38)).
For the weight (x) = 1 of the Fibonacci case, we have from (39) the following alternating sum of squares of Fibonacci polynomials

!
. When x = s = 1, the weights (37) are 3± p 5 2 .In this case we have the following numerical formulas for weighted sums of squares of Fibonacci and Lucas numbers in terms of Fibonomials: By using the weights (37) in ( 39) and (40), we can obtain expressions for weighted sums of squares of Fibonacci and Lucas polynomials, in which the weight functions are in turn certain Fibonacci or Lucas polynomials.This is the content of the following proposition.Proposition 3.For l 2 Z we have the following weighted sums of squares of Fibonacci and Lucas polynomials (a) . (b) .
Proof.First write the sum (39) as , where l 2 Z. Substitute the weights (37) in ( 46), take the di↵erence of the resulting expressions, multiply both sides of this di↵erence by x 2 + 4 .
Finally, use the identity to obtain (42).Similarly, if we substitute the weights (37) in (46), then take the sum of the resulting expressions, then use the Binet's formula L r (x) = ↵ r (x) + r (x), and then use the identity we obtain (43).In a similar fashion, if we begin now with (40), written as

◆
Fs(x,y)= F sn (x, y) F s(n 1) (x, y) • • • F s(n p+1) (x, y) F s (x, y) F 2s (x, y) • • • F ps (x, y) , 0 < p < n.If n or p are negative, or p > n, we have n p Fs(x,y) = 0.It is known that n p Fs(x,y) are indeed polynomials in x and y.When x = y = 1 we have the s-Fibonomials n p Fs